Linear Graph Transformations on Spaces of Analytic Functions

Let H be a Hilbert space of analytic functions with multiplier algebra M(H), and let M = {(f, T1f...., Tn−1f) : f ∈ D} be an invariant graph subspace for M(H). Here n ≥ 2, D ⊆ H is a vector-subspace, Ti : D → H are linear transformations that commute with each multiplication operator Mφ ∈ M(H), and M is closed in H. In this paper we investigate the existence of nontrivial common invariant subspaces of operator algebras of the type AM = {A ∈ B(H) : AD ⊆ D : ATif = TiAf ∀f ∈ D}. In particular, for the Bergman space La we exhibit examples of invariant graph subspaces of fiber dimension 2 such that AM does not have any nontrivial invariant subspaces that are defined by linear relations of the graph transformations for M.